Let \(T\) be a bounded linear operator acting on a separable infinite dimensional Hilbert space such that \(T\) is quasitriangular and the spectrum of \(T\) is connected. It is shown that for any \(\varepsilon > 0\) there exists a compact operator \(K\) with \(\|K\|< \varepsilon\) such that \(T+K\) is strongly irreducible. This result gives a partial answer to a question posed by D. A. Herrero.